Finding the Association of a Risk Factor in Medical Research

Association between a risk factor (exposure) and a disease (outcome) is measured using a combination of study design, statistical measures, and causal reasoning. Here is the full framework:

1. The 2x2 Contingency Table - The Foundation

All association measures come from this fundamental table:

2x2 Table: Calculating Rates and Measures

	Outcome +	Outcome -
Exposure +	a	b
Exposure -	c	d

a = exposed WITH disease
b = exposed WITHOUT disease
c = unexposed WITH disease
d = unexposed WITHOUT disease

2. Choosing the Right Study Design

The study design determines which measures of association you can calculate:

Study Type	How It Works	Best For	Association Measure
Cohort	Identify exposed/unexposed, follow forward for outcome	Common outcomes, temporal relationship	Relative Risk (RR)
Case-Control	Identify cases/controls, look back at exposure	Rare diseases	Odds Ratio (OR)
Cross-Sectional	Measure exposure + outcome at same time	Prevalence, descriptive data	Prevalence Ratio
RCT	Randomly assign exposure/intervention	Causal inference	RR, ARR, NNT

"Observational studies, including cohort, case-control, and cross-sectional studies, are analytic studies that take advantage of 'natural experiments' in which exposure is not assigned by the investigator." - Berek & Novak's Gynecology, p. 120

3. Key Measures of Association

Relative Risk (RR)

Formula: a/(a+b) ÷ c/(c+d)
Used in: Cohort studies and RCTs (where total population is known)
Interpretation:
- RR = 1: no association
- RR > 1: exposure increases risk
- RR < 1: exposure is protective
Example: RR = 1.6 means exposed individuals are 1.6 times as likely to develop the disease

Odds Ratio (OR)

Formula: (a/b) ÷ (c/d) = ad/bc
Used in: Case-control studies primarily
Note: OR approximates RR when the outcome is rare (rare disease assumption)
OR > 1 = greater odds of disease in exposed group; OR < 1 = lower odds

Attributable Risk (AR) - also called Risk Difference

Formula: a/(a+b) - c/(c+d)
Answers: "How much greater is the absolute risk in the exposed group?"
"Attributable risk is not directly obtainable in a case-control study." - Berek & Novak's Gynecology, p. 125

Attributable Risk Percent (ARP)

Formula: AR / Risk in exposed group
Answers: "What percentage of disease in exposed individuals is attributable to the exposure?"

Absolute Risk Reduction (ARR)

Formula: Risk in control group - Risk in intervention group
Used when the exposure is protective/therapeutic

Number Needed to Harm (NNH) / Number Needed to Treat (NNT)

Formula: 1 / ARR
NNT: how many need treatment to prevent one outcome
NNH: how many exposures before one extra case of harm

4. Statistical Significance

Once an association measure is calculated, you need to determine if it is statistically real or could be due to chance:

p-value

Probability the observed result occurred by chance alone under the null hypothesis
Conventional threshold: p < 0.05 (5% chance the association is spurious)
For genome-wide association studies (GWAS), this is set much lower (p < 10^-7) because hundreds of thousands of variants are tested

Confidence Intervals (CI)

A 95% CI means: if the study were repeated many times, 95% of the calculated CIs would contain the true population value
For OR or RR: if the 95% CI includes 1.0, the association is not statistically significant
Narrow CI = more precision; wide CI = less precision (often from small sample size)

5. Controlling for Bias and Confounding

Association does not equal causation. You must account for:

Bias Types

Selection bias: systematic error in who is included (e.g., controls are not representative)
Information/misclassification bias: incorrect classification of exposure or disease
Recall bias: cases remember past exposures better than controls - especially a problem in case-control studies

Confounding

A confounder is a variable associated with both the exposure AND the outcome. It can create a false association or mask a real one.

How to control confounding:

Randomization (RCTs) - distributes confounders equally
Restriction - limit study to one level of the confounder
Matching - pair cases and controls on confounder variables
Stratification - analyze within subgroups
Multivariable analysis - statistically adjust for multiple confounders simultaneously, producing an adjusted OR or RR

"Multivariable analysis is a statistical technique commonly used in epidemiologic studies that simultaneously controls a number of confounding variables." - Berek & Novak's Gynecology, p. 130

6. From Association to Causation - Bradford Hill Criteria

Finding a statistical association is not enough to conclude causation. Bradford Hill's criteria help judge causality:

Criterion	What It Means
Strength	The further RR/OR from 1.0, the more likely it's real
Temporality	Exposure must precede disease (most critical criterion)
Dose-response	Higher exposure = higher risk (biologic gradient)
Consistency	Association replicated by different investigators/settings
Plausibility	A biologic mechanism can explain the association
Coherence	Compatible with known biology of the disease
Experiment	Disease reduced when exposure is removed

7. Worked Example (Smoking and Low Birth Weight)

	Low Birth Weight	Normal Birth Weight
Smokers	40	311
Non-smokers	331	4318

Risk in smokers: 40/351 = 11.4%
Risk in non-smokers: 331/4649 = 7.1%
RR = 0.114/0.071 = 1.6 (smokers 1.6x more likely to have low-birth-weight baby)
OR = (40 × 4318)/(331 × 311) = 1.68
Attributable Risk = 0.114 - 0.071 = 0.043 (4.3 extra cases per 100 smokers)
ARP = 0.043/0.114 = 38% of low birth weight in smokers is attributable to smoking

Summary: Which Measure to Use When

Situation	Use
Cohort or RCT	Relative Risk (RR), Attributable Risk
Case-control	Odds Ratio (OR)
Therapeutic intervention	ARR, NNT, RRR
Harmful exposure	NNH, Attributable Risk %
Rare disease in any design	OR approximates RR
Multiple confounders	Adjusted OR/RR from multivariable regression

The gold standard for establishing causation remains the randomized controlled trial, but observational studies with strong methodology, large sample sizes, and careful confounding control can provide compelling evidence of association. - Berek & Novak's Gynecology, p. 117-131

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